Additionally, we argue that both spin-orbit interaction terms in \({\cal H}^{\prime }\)  inevitably affect the band characteristics or even the electronic topological properties [1]}, [2]}, [3]}, [4]}. In our latest theoretical work [5]}, the analytical processes related to spin-orbit coupling and magnetism in BLG are discussed in detail (due to the complexity of the expressions, we do not show them here). In particular, the sign of \(\nu \)  and \(\chi \)  in \(\cal H^{\prime }\)  is more likely to be negative [5]}, [7]}, [4]} due to structural symmetry in real doubly-proximitized BLG samples.
